Question

Find the solution to the differential equation dy/dx = cos(x) / y2 , where y(π/2) = 0.

Answer 1

`y(x)=\sqrt[3]{sin(x)-3}`

Step-by-step explanation:

The differential equation is:

`(dy)/(dx)=(cos(x))/(y^2)`

Separate variables:

`y^2dy=cos(x)dx`

Integrate both sides:

`\int y^2dy= \int xcos(x)dx`

`(y^3)/(3)=sen(x)+C`

`y^3=3sin(x)+C'`

Find C' using the inital condtion y(π/2) = 0

`0=3sin(\pi /2)+C'\\\\0=3+C'\\ \\ C'=-3`

Then,

`y^3=3sin(x)-3\\ \\ \\ y=\sqrt[3]{sin(x)-3}\\ \\ \\ y(x)=\sqrt[3]{sin(x)-3}`